ZeroTruncatedGeometric: The Zero-Truncated Geometric Distribution

In actuar: Actuarial Functions and Heavy Tailed Distributions

The Zero-Truncated Geometric Distribution

Description

Density function, distribution function, quantile function and random generation for the Zero-Truncated Geometric distribution with parameter prob.

Usage

dztgeom(x, prob, log = FALSE)
pztgeom(q, prob, lower.tail = TRUE, log.p = FALSE)
qztgeom(p, prob, lower.tail = TRUE, log.p = FALSE)
rztgeom(n, prob)

Arguments

x	vector of (strictly positive integer) quantiles.
q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
prob	parameter. 0 < prob <= 1.
log, log.p	logical; if TRUE, probabilities p are returned as \log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X \le x], otherwise, P[X > x].

Details

The zero-truncated geometric distribution with prob = p has probability mass function

% p(x) = p (1-p)^{x - 1}

for x = 1, 2, \ldots and 0 < p < 1, and p(1) = 1 when p = 1. The cumulative distribution function is

P(x) = \frac{F(x) - F(0)}{1 - F(0)},

where F(x) is the distribution function of the standard geometric.

The mean is 1/p and the variance is (1-p)/p^2.

In the terminology of Klugman et al. (2012), the zero-truncated geometric is a member of the (a, b, 1) class of distributions with a = 1-p and b = 0.

If an element of x is not integer, the result of dztgeom is zero, with a warning.

The quantile is defined as the smallest value x such that P(x) \ge p, where P is the distribution function.

Value

dztgeom gives the (log) probability mass function, pztgeom gives the (log) distribution function, qztgeom gives the quantile function, and rztgeom generates random deviates.

Invalid prob will result in return value NaN, with a warning.

The length of the result is determined by n for rztgeom, and is the maximum of the lengths of the numerical arguments for the other functions.

Note

Functions {d,p,q}ztgeom use {d,p,q}geom for all but the trivial input values and p(0).

rztgeom uses the simple inversion algorithm suggested by Peter Dalgaard on the r-help mailing list on 1 May 2005 (https://stat.ethz.ch/pipermail/r-help/2005-May/070680.html).

Author(s)

Vincent Goulet [e-mail]

References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

See Also

dgeom for the geometric distribution.

dztnbinom for the zero-truncated negative binomial, of which the zero-truncated geometric is a special case.

Examples

p <- 1/(1 + 0.5)
dztgeom(c(1, 2, 3), prob = p)
dgeom(c(1, 2, 3), p)/pgeom(0, p, lower = FALSE) # same
dgeom(c(1, 2, 3) - 1, p)                        # same

pztgeom(1, prob = 1)        # point mass at 1

qztgeom(pztgeom(1:10, 0.3), 0.3)

actuar documentation built on Nov. 8, 2023, 9:06 a.m.